Electron. J. Diff. Equ., Vol. 2013 (2013), No. 182, pp. 1-16.

Reconstructing the potential function for indefinite Sturm-Liouville problems using infinite product forms

Mohammad Dehghan, Ali Asghar Jodayree

Abstract:
In this article we consider the linear second-order equation of Sturm-Liouville type
$$ y''+(\lambda\phi^2(t)-q(t))y=0, \quad 0\leq t\leq 1, $$
where $\lambda$ is a real parameter, $q(t)$ is the potential function and $\phi^2(t)$ is the weight function. We use the infinite product representation of the derivative of the solution to the differential equation with Dirichlet-Neumann conditions, and for the system of dual equations which is needed for expressing inverse problem and for retrieving potential. It must be mentioned that the weight function has a zero whose order is an integer called a turning point.

Submitted July 15, 2013. Published August 7, 2013.
Math Subject Classifications: 34B24, 34A55, 34E20, 34E05.
Key Words: Indefinite Sturm-Liouville problem; turning point; dual equations; infinite product form.

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Mohammad Dehghan
Faculty of Mathematical Sciences
University of Tabriz
Tabriz, Iran
email: m-dehghan@tabrizu.ac.ir, Tel. +989113512619
Ali Asghar Jodayree
Faculty of Mathematical Sciences
University of Tabriz
Tabriz, Iran
email: akbarfam@yahoo.com

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